Capacity Bounds for Ad hoc and Hybrid Wireless Networks

Transcription

1 Capacity Bouds fo Ad hoc ad Hybid Wieless Netwoks Ashish Agawal Depatmet of Compute Sciece, Uivesity of Illiois 308 West Mai St., Ubaa, IL , USA. P. R. Kuma Coodiated Sciece Laboatoy, Uivesity of Illiois, 308 West Mai St., Ubaa, IL , USA. Web: pkuma ABSTRACT We study the capacity of static wieless etwoks, both ad hoc ad hybid, ude the Potocol ad Physical Models of commuicatio, poposed i []. Fo ad hoc etwoks with odes, we show that ude the Physical Model, whee sigal powe is assumed to atteuate as /, >, the taspot capacity scales as Θ( bit-metes/sec. The same boud holds eve whe the odes ae allowed to appoach abitaily close to each othe ad eve ude a moe geealized otio of the Physical Model wheei the data ate is Shao s logaithmic fuctio of the SINR at the eceive. This esult is shap sice it closes the gap that existed betwee the pevious best kow uppe boud of O( ad lowe boud of Ω(. We also show that ay spatio-tempoal schedulig of tasmissios ad thei ages that is feasible ude the Potocol Model ca also be ealized ude the Physical Model by a appopiate choice of powe levels fo appopiate thesholds. This allows the geealizatio of vaious lowe boud costuctios fom the Potocol Model to the Physical Model. I paticula, this povides a bette lowe boud o the best case taspot capacity tha i []. Fo hybid etwoks, we coside a ovelay of µ adomly placed wied base statios. It has peviously bee show i [6] that if all odes adopt a commo powe level, the each This mateial is based upo wok patially suppoted by AFOSR ude Cotact No. F , USARO ude Cotact Nos. DAAD ad DAAD , AFOSR ude Cotact No. F , ad NSF ude Cotact Nos. NSF ANI ad CCR , ad DARPA ude Cotact Nos. N ad F C-905. Coespodig autho ode ca be povided a thoughput of at most Θ( to log adomly chose destiatios. Hee we show that by allowig odes to pefom powe cotol ad popely choosig µ, it is futhe possible to povide a thoughput of Θ( to ay factio f, 0 < f <, of odes. This esult holds ude both the Potocol ad Physical models of commuicatio. O the oe had, it shows that that the aggegate thoughput capacity, measued as the sum of idividual thoughputs, ca scale liealy i the umbe of odes. O the othe had, the esult udescoes the impotace of choosig miimum powe levels fo commuicatio ad suggests that simply commuicatig with the closest ode o base statio could yield good capacity eve fo multihop hybid wieless etwoks. Keywods Wieless etwoks, ad hoc etwoks, hybid etwoks, physical model, potocol model, thoughput capacity, taspot capacity.. INTRODUCTION Asymptotic bouds o the capacity of ad hoc wieless etwoks have bee established i []. Two models of commuicatio wee poposed. The Potocol Model equies that each eceive lie outside the itefeece egio of evey othe tasmitte. Hee each tasmitte tasmittig to a eceive at distace foms a itefeece egio cosistig of a disc of adius (+ ceteed aoud itself. Ude the alteate Physical Model of commuicatio, the sigal to itefeece plus oise atio (SINR is equied to be above a pe-specified theshold β, whee the sigal powe is assumed to decay as with distace, fo >. Futhe, two metics fo capacity ae also defied i []. Taspot capacity is defied as the total bit-distace poduct pe secod that ca be taspoted by the etwok. Thoughput capacity, o the othe had, is defied as the maximum commo thoughput that ca be povided to each ode with a adomly chose destiatio. Ude the Potocol Model, uppe ad lowe bouds i bit- 8 W + metes/sec of W ad ae obtaied fo the best case taspot capacity i [], whee W is the capacity of the wieless chael i bits/sec. Fo adom etwoks, i.e., etwoks whee the ode ae dis- + 8, espectively, ACM SIGCOMM Compute Commuicatios Review 7

2 tibuted uifomly ad idepedetly, ad a destiatio is adomly chose fo each ode, the uppe boud o the pe c ode thoughput capacity is show to be W ude log the Potocol Model, fo some costat c. Fo the Physical Model, a uppe boud of O( o the taspot capacity has bee established i []. The same boud is obtaied i [7] fo a Geealized Physical Model, whee data ate is the Shao logaithmic fuctio of the SINR. Howeve the best lowe boud obtaied has bee Ω(, leavig a gap betwee the uppe ad the lowe bouds. Ifomatio theoetic appoaches have also bee used to detemie scalig laws o the total taspot capacity of the etwok. I [3], assumig that the iteode distace is lowe bouded, ad > 6, it is show that the taspot capacity is asymptotically bouded by the sum of the tasmit powe of the odes i the etwok. Thus fo domais of size Θ(, taspot capacity scales as Θ(. This esult applies to the Physical Model too. Howeve, it is ot kow if the same boud holds i geeal fo >, o whe the odes ae allowed to appoach abitaily close to oe aothe. Fo the case of the Potocol Model, it has bee show i [] that discs of adius aoud the eceives ae exclusio egios, i.e., ae mutually disjoit fom each othe, whee is the distace fom the tasmitte to the paticula eceive. I [], lage exclusio egios have bee obtaied, thus impovig the capacity bouds fo the Potocol Model. A uppe boud of, ad also a impoved lowe boud of 8 W (+ + W (+ + bit-metes/sec ae obtaied. Thus the best achievable taspot capacity has bee backeted to withi a facto of 8.83, iespective of, chaacteizig it faily shaply. I this pape, we fist show that Potocol Model is a moe estictive model i compaiso to the Physical Model i the followig sese. Specifically, we pove that cofiguatios satisfyig the costaits of the potocol model fo suitable choices of also satisfy the costaits of a appopiate physical model. This immediately establishes that lowe boud costuctios ude the potocol model cotiue to hold ude the physical model. I paticula, the gid of ellipses costuctio fo the Potocol Model i [] woks fo the Physical Model also. This povides a shape lowe boud o the best kow taspot capacity ude Physical Model. Futhe, we impove the uppe boud fo the Physical Model to Θ(, thus establishig a shap ode of the taspot capacity. The same boud holds eve fo the Geealized Physical Model metioed befoe. Besides, ou poof does ot impose ay lowe bouds o the iteode distace, thus allowig the esult to hold fo both dese ad spase etwoks. Fo a etwok with odes, if the domai size is Θ(, the uppe boud o taspot capacity is Θ( fo ay > ad with the costait that the maximum powe level is bouded by Θ(. Also the capacity is achievable by a simple gid aagemet of odes. Thus fo the special case of the Geealized Physical Model, we ae able to chaacteize the scalig law fo all >, ulike the ifomatio theoetic settig [3]. I [4], it has ecetly bee show that the maximum commo thoughput that ca be fuished is Θ( whe powe cotol ca be execised with some odes choosig lowe powe ad some highe. Ou best case uppe boud of Θ( thus establishes that adom etwoks ae best case up to ode. Next we exploe the thoughput capacity of hybid wieless etwoks. Fo the case of ad hoc etwoks, it is show i [] that thoughput capacity scales as Θ( log, which teds to zeo as the umbe of odes iceases. I [5], it is show that with wieless odes, the umbe of base statios must scale as at least Θ( to achieve bette scalig of capacity tha pue ad hoc etwoks. I [6], it is show that fo etwoks whee the atio of wieless odes to base statios is bouded above by some costat, a thoughput capacity of Θ( is the optimal achievable. A impotat log assumptio hee is that all odes choose a commo powe level. We show that by employig powe cotol i the etwok, oe ca futhe impove the thoughput capacity. Fo a adom etwok with wieless odes ad µ base statios, µ > 0, we show that oe ca additioally povide a thoughput of Θ( to some Θ( odes with pobability appoachig as. This esult shows that the total capacity of adom hybid etwoks, measued as the sum of the thoughputs fuished to the odes, ca scale liealy with the umbe of odes. Futhe ay factio f, 0 < f <, of odes ca be fuished this thoughput povided that µ is sufficietly lage. Also the scheme which achieves this equies odes to commuicate with thei closest base statios thus savig battey powe ad udescoig the impotace of usig miimum powe levels fo commuicatio. Some of the implicatios fo the desig ad opeatio of wieless etwoks ae the followig. Much cuet techology fo decodig packets is essetially based o the Physical Model, whee itefeece is teated as oise, athe tha o usig sophisticated multi-use detectio whee multiple itefeig sigals ae simultaeously decoded at a ode. Fo all wieless commuicatio etwoks fomed by usig such techology, the peset pape chaacteizes shaply the ode of what is achievable. A class of applicatio sceaios of much cuet iteest is seso etwoks. Thee, at peset the emphasis is o usig vey low cost odes whee, cuetly at least, sophisticated multi-use detectio is ot evisaged. By allowig odes to be abitaily closely placed, we attempt to captue deploymets of high desity, as fo example i seso etwoks as well as closely spaced pesoal aea etwoks. At the othe ed, fo hybid etwoks, ou esults udelie the impotace of buildig powe cotol ito the potocol stack. By usig powe cotol, we show that the thoughput fuished to ay factio of uses ca be impoved i ode. Moeove, sice the stategy achievig this cosists of commuicatig with eaby odes ad usig them fo elayig, it suggests that potocols should be desiged to be efficiet i the egime whee the umbe of hops is lage. Thus, attempts should be made to educe all pe hop costs. This will become iceasigly impotat fo lage scale deploymets of wieless etwoks as well as ACM SIGCOMM Compute Commuicatios Review 7

3 hybid etwoks.. NETWORK MODEL Coside a etwok of odes i a disc of uit aea. Let X i, i, deote the locatio of ode i. We will use X k to deote a ode as well as its locatio. Let {(X k, X R(k : k T } be the set of all active tasmitte-eceive pais i some paticula slot. As i [], we assume a slotted model fo coveiece of expositio. Let the tasmissio adius, X k X R(k, be deoted as k. We fist descibe the models of commuicatio.. The Potocol Model The tasmissio fom ode X i, i T, is successfully eceived by the eceive X R(i oly if X k X R(i ( + X k X R(k, ( fo evey k T \i. Hee > 0 models a guad zoe aoud the tasmissio egio. Equivaletly, oe ca coside a guad zoe aoud the eceive as i []. Geealized Potocol Model As befoe, let {(X k, X R(k : k T } deote the set of all active tasmitte eceive pais. Associated with each tasmitte-eceive pai, (X k, X R(k, k T, let thee be a itefeece egio, I k, ad assume that X R(k I k. Suppose also that the ecessay coditio fo a tasmissio fom X k to X R(k to be successful is X R(k / I j, j T, j k. ( Also o ode ca seve simultaeously as a eceive as well as a tasmitte. Such a geeal itefeece footpit I k ca be used to model, fo example, diectioal ateas.. The Physical Model Suppose sigal powe suffes a atteuatio by a facto of as it taveses a distaces, whee > epesets the path loss expoet. The the eceived sigal-to-itefeece-plusoise atio (SINR fo the tasmissio fom X k to X R(k is X k X R(k N 0 + i T,i k, (3 X i X R(k whee N 0 epesets the ambiet oise. Ude the physical model of commuicatio, the tasmissio fom X k to X R(k is successful oly if its SINR is geate tha β. Geealized Physical Model The Physical Model assumes a theshold based chael, whee the sigal ca be decoded at a costat specified ate W bits/sec oly if the SINR is geate tha some theshold, failig which o thoughput is eceived at all. Thus the coespodig thoughput is eithe 0 o W bits/s. Hee we geealize this otio as i [7] ad assume that the thoughput is a fuctio of the SINR at the eceive. We use Shao s capacity fomula fo the additive white Gaussia oise chael [8]. Thus the data ate fo the tasmitte eceive pai (X k, X R(k is give i bits/sec by X k X R(k W k H m log ( + N 0H m + i T,i k (4 X i X R(k whee H m is the badwidth of chael m i hetz, ad N 0 is the oise spectal desity i watts/hetz. 3. CAPACITY BOUNDS FOR AD HOC WIRE- LESS NETWORKS UNDER THE PRO- TOCOL MODEL I [], uppe bouds o the best case taspot capacity ude the Potocol Model ae deived. It uses the fact that the eighbohood aoud a eceive of a tasmissio of age is a exclusio egio. Fomally, a exclusio egio is a aea associated with a active eceive that must emai disjoit fom the exclusio egio of evey othe active eceive i the etwok at that time, fo ay cofiguatio of tasmittes ad eceives. I [], it is show that these exclusio egios ae much lage, ad ca be computed fo abitay itefeece footpits icludig those aisig fom diectioal ateas. Specifically, fo the Geealized Potocol Model, Theoem. i [] states that the set E k {P : P X R(k < P Q, Q / I k } is a valid exclusio egio fo each eceive R(k. Futhe, Theoem 3.3 states this set is always covex. Fo the case of Potocol Model, i.e., whee I k is a disk of adius ( + X i X R(i ceteed at X i, the set E k coespods to a ellipse with X k ad X R(k as the foci, ad ecceticity, i.e., E + k {P : P X R(k + P X k ( + X R(k X k }. Usig these exclusio egios, a uppe boud i bit-metes/sec of 8 W (+ + is deived fo the best case taspot capacity. Futhe a simple gid of ellipses cofigu- allows a lowe boud costuctio which achieves atio bit-metes/sec. Thus, caefully cha- W (+ + acteizig the exclusio egios allows backetig the best case taspot capacity fo Potocol Model to withi a facto of 8. Similaly, fo adom etwoks whee ode locatios ae chose uifomly ad idepedetly, it is show that the thoughput capacity is bouded by whee c is some costat. c W (+ + log, 4. CAPACITY BOUNDS FOR AD HOC WIRE- LESS NETWORKS UNDER THE PHYSI- CAL MODEL I this sectio, we boud the best case taspot capacity of wieless etwoks ude the two Physical Models discussed i Sectio. We begi by showig how the Potocol Model compaes with these two models. 4. Physical Model Compaed with Potocol Model The costaits defied by the Potocol Model ae local. They oly equie cetai localities of tasmittes to be fee of eceives. O the othe had, the Physical Model cosides the cumulative itefeece due to all the odes i the etwok. Thus, ituitively it appeas that the Physical Model is a much moe estictive model, ad would etail lowe capacity. Howeve, cotay to ituitio, it tus out that it is i fact the Potocol Model that ca be cosideed as the moe estictive model, as the ext theoem shows. ACM SIGCOMM Compute Commuicatios Review 73

4 Theoem 4.. Give ay β > 0, thee is a β > 0, such that fo ay > β, if ay cofiguatio {(X k, X R(k, k T } satisfies the Potocol Model with guad paamete, the thee is a assigmet of powes, {, k T }, such that the cofiguatio also satisfies the costaits of the Physical Model with SINR theshold β. I paticula, β (c β /, whee c : 4. Poof: Fom [], we kow that the exclusio egio fo the pai (X k, X R(k is a egio bouded by the followig ellipse: Now the SINR fo the pai (X k, X R(k is k N 0 + I k c k N 0 + cc ( k N 0 c k + c (sice k c β. E k {P : P X R(k + P X k ( + X R(k X k }. I paticula, let D k ad D k deote disks of adius k aoud X k ad X R(k espectively. The both these disks ae cotaied i the ellipse. Thus, fo ay distict i ad j, D i is mutually disjoit with D j ad D j. Let the ambiet oise be N 0. We will show that a powe assigmet that woks is c k, whee c N 0 c (. To show this, we fist boud the itefeece at ay eceive due to all the othe tasmittes. Let D ij : {x : x D i, x X R(j X i X R(j }. Thus D ij epesets the pat of D i which is close to the eceive X R(j tha the tasmitte X i. Coside a cicle ceteed at X R(j ad passig though X i, ad let it cut the bouday of D i at A ad B. The, sice X R(j X i X R(j A A X i, AX ix R(j /3. Similaly, BX ix R(j /3, Thus the aea of D ij is at least oe thid of the aea of D i. Let I k deote the itefeece at eceive X R(k. We ow have, I k : i T,i k i T,i k i T,i k i T,i k X i X R(k 4c da X i X R(k S k c c D i D ik D ik da X i X R(k da x X R(k (whee x is the positio vecto of elemet da c da x X R(k c (whee S k : i T,i k D ik c da x X R(k x X R(k k / (sice S k D k φ k / d 4c ( k cc ( k. Thus we have show that ay spatio-tempoal schedulig, alog with choices of ages satisfyig the costaits of the Potocol Model, also admits a choice of powe levels which togethe with appopiate thesholds also satisfy the costaits of the Physical Model. This allows diect geealizatio of vaious lowe boud costuctios fo the Potocol Model to the Physical Model. I paticula, by exploitig the gid of ellipses cofiguatio i [], which yields the lagest lowe boud o the best case taspot capacity kow to date (as fa as the authos ae awae fo the Potocol Model, we get the followig lowe boud esult fo the Physical Model. Note that it shapes the lowe boud fo taspot capacity ude the Physical Model obtaied i []. Coollay 4.. Thee is a placemet of odes iside a uit disk ad a assigmet of taffic pattes, such that, ude the Physical Model with SINR theshold β, the etwok ca achieve whee : (c β /. W ( + + bit-metes/sec, (5 4. A Shap Uppe Boud o the Taspot Capacity ude the Geealized Physical Model I this sectio, we will establish uppe bouds o the best case taspot capacity of ad hoc wieless etwoks ude the Geealized Physical Model descibed i Sectio. Note that, asymptotically, the same boud holds fo the Physical Model as well. To see this, ote that fo ay give cofiguatio of tasmitte-eceive pais, togethe with the choice of tasmissio powes, a paticula eceive would eceive W bits/sec of data ude the Physical Model oly whe the SINR is geate tha some theshold β. I that case, the Geealized Physical Model would also allow some costat thoughput depedig upo β. Thus, withi a costat, the total bit-distace/sec obtaied ude the Geealized Physical Model would be at least as much as that obtaied ude the Physical Model. Tag maximum ove all possible cofiguatios, the taspot capacity ude the Geealized Physical Model is asymptotically lowe bouded by that ude the Physical Model. ACM SIGCOMM Compute Commuicatios Review 74

5 We ext show the shap esult that the boud is Θ(, which is asymptotically the same as the lowe boud i Coollay 4.. This diectly impoves the pevious boud of i [], ad povides a uppe boud which coicides i ode with the lowe boud, thus shaply chaacteizig the scalig law. We fist establish some esults that ae used to pove the mai esult. m, we get f(m 4f(m/ + m 4(3 m 4 m + m 3 m m. We ae ow eady to peset ou mai esult. Lemma 4.. Fo x R + ad, l ( + x x. Poof: l ( + x l (( + x l ( + x x. Lemma 4.. Coside a m m squae gid with each gid box beig a uit squae. Let m be a powe of. Let S {Q i, i k} be a set of k poits with k m, such that each squae i the gid cotais at most oe poit fom S. Let thee be a total odeig imposed o the k poits i S. Fo i k, defie d i mi({m } { Q j Q i : j k, j i, Q i Q j}, i.e., the distace to the eaest highe odeed poit. The k i di 3 m m. Poof: Let f(m deote the uppe boud o k i di ove ay choice of the poits ad thei odeig. We will obtai a ecuece fo f(m. Note fist that f(. Fo m 4, divide the gid ito fou equal quadats of size m m. Fo ay poit Q i, let d i mi({ m } { Qj Q i : j k, j i, Q i ad Q j ae i the same quadat, Q i Q j} be similaly defied with espect to the quadat i which the poit lies. Next ote that fo ay two poits Q i ad Q j i the same quadat, if Q i Q j, the d i Q i Q j m, i.e., at least oe of them has d i m. Thus, thee ca be at most oe poit i each quadat with d i m. Such a poit ca oly be the maximal elemet i the quadat udethe odeig. Fo such a maximal elemet, d i m. Also, if Q j is ot the maximal elemet i the quadat, the d j d j sice d j ivolves tag miimum ove a lage set. We ow have k i di k i d i + 4(m m. Tag maximum ove all cofiguatios, we get f(m 4f(m/ + m. The base coditio is f(. We ca show by iductio that the solutio of this ecuece is f(m (3 m m. Fo m, the base coditio is satisfied. Fo Theoem 4.. Coside the Geealized Physical Model with >, oise spectal desity N 0 /, ad available badwidth H hetz, that ca be shaed by M sub-chaels, m,..., M with badwidths H m satisfyig M m H m H. Suppose that the maximum powe level P max used ove a subchael m with badwidth H m hetz is bouded by P max H m N 0, whee is the umbe of odes i a squae of uit aea. The the taspot capacity, fo ay abitay placemet of odes ad choice of tasmissio powes, is uppe bouded by Hk bit-metes/sec, (6 whee k log (e(( (( +. Poof: We will assume a slotted opeatio with slot legth τ secs. As ealie, let {(X k, X R(k : k T } be the set of all active tasmitte-eceive pais. Let k : X k X R(k ad : X k X R(i. Let be the powe level used by ode i, ad let w be a ode tasmittig at powe level max i T { }. Let T T \{w}. The the total taffic i bit-metes fo slot s ad chael m, with W i deotig the data ate obtaied by the i th tasmitte, is give by: i W i τ i T i i τh m log ( + N i T 0 H m + k T,k i w τh m log ( + i T i τh m log ( + P w w N 0 H m + P w i k T,k i τh m log (e(( / + N 0 H m i T i l ( + i k T,k i τh m log (e( + i T i l ( + k i (usig Lemma 4. i k T,k i (usig P w P max N 0 H m. (7 We will ow compute bouds o the sum i (7. Fo this, we fist divide T ito disjoit classes C 0, C,... as follows. Fo j 0, defie R j : j log (. Note that j < R j j. Let C 0 : {i : i T, X i X R(i < R 0 }. Fo j > 0, defie C j : {i : i T, R j X i X R(i < R j }. ACM SIGCOMM Compute Commuicatios Review 75

6 Note that we ca wite the sum tem o the RHS of (7 as i T i l ( + i l ( + j 0 i C j i k T,k i We will ow detemie i C j i l ( + i k T,k i i k T,k i fo each j 0. To do this, we divide ou squae domai ito a squae gid with gid legth R j. Let us suppose that thee ae gid squaes each with at least oe eceive, ad let us deote them by Γ g, g, whee. Deote the umbe of tasmitte-eceive pais whose R j eceive lies i cell Γ g by g. Let the tasmitte-eceive distaces ad tasmitted powe levels fo these pais be deoted espectively by gh ad P gh, whee 0 h g. Let the total itefeece at the eceive be deoted by I gh. Fo each Γ g, let Q g deote the eceive coespodig to the pai with the highest tasmitted powe level amog all pais with eceives i Γ g. Assume that g0, ad I g0 coespod to this eceive Q g. The, i l ( + i C j g0 l ( + g g g h i k T,k i g0 I g0 + gh l ( + P gh gh I gh. (8 Next coside all the maximum powe eceives Q g, g. Defie a total odeig by odeig them accodig to thei tasmitted powe levels ad beag ties abitaily (so < P h0 Q g Q h. Defie d g mi({ } { Q h Q g : h, h g, Q g Q h }. Fist ote that the ode w, which is the tasmitte usig the globally maximum powe level, is withi a distace of evey Q g, g. Thus, fom the defiitio of d g, fo ay eceive Q g, thee must be a tasmitte withi a distace d g of Q g that is tasmittig with powe at least. Thus, the itefeece I g0 is at least I g0 (d g + R j. (9 Similaly, we obtai a boud o I gh, h g. Coespodig to the gid box Γ g, defie P g : g h P gh. (0 The I gh Pg P gh (, by cosideig oly those tasmittes as itefees whose eceives ae withi the gid squae R j +R j Γ g, ad uppe boudig the distace of such tasmittes. Sice fo h g, P gh, we have, P g P gh, o equivaletly, P g P gh Pg. Thus, the itefeece I gh is bouded as I gh P g/ ( R j + R j. ( We will ow use (9 ad ( to simplify (8. Fist, we have g0 l ( + g g0 l ( + g g0 I g0 g0 (usig (9 (d g +R j (d g + R j (usig Lemma 4. g (R j (3 R j + R j( (usig Lemma 4. R j (3 +. ( Next ote that we have g g h g g h gh l ( + Fo j 0, (3 simplifies as, G 0 g g h G 0 g g h P gh gh I gh gh l ( + P gh (( + Rj. (3 P g gh gh l ( + P gh (( + R0 P g gh gh l ( + (( + R 0 gh (sice P g P gh G 0 g g h ( G 0 + R 0 ( + R 0 (usig Lemma 4. 0 g h ( + ( ( +. (4 ACM SIGCOMM Compute Commuicatios Review 76

7 Fo j > 0, (3 simplifies as, g g h g g h gh l ( + P gh (( + Rj P g gh R j l ( + P gh (( + Rj P g R j (sice R j gh < R j g g h R j l ( + P gh P g (( + (sice R j R j (( + R j g g h (sice l( + x x P gh P g (( + R j (usig (0 g (( + R j( (sice Gj R R j j (( +. (5 R j Fially, we ca simplify the summatio i (7 as follows: i l ( + i T i k T,k i i l ( + j 0 i C j j 0 g0 l ( + g g j 0 g h j 0 i k T,k i g0 I g0 + gh l ( + P gh gh I gh R j (3 + + (( + + ((( + (usig (, (4 ad (5 R j j>0 R 0 (3 + j 0 j + (( + + ((( + j (sice R j j R 0 R 0 j>0 < (3 + ( + (( + + ((( + 4 ( (sice R j > j (( (( +. (6 Usig (6, we ow simplify (7. Thus, i W i τ i T τh m log (e( + (( (( + τh m log (e (( (( + τh m k. Now assume that the tasmissio lasts fo a total time of S sec. The total capacity i bit-metes/sec is the, S/τ S S M i W i τ s m i τ S/τ s m S/τ τk S M (τh m k M s m S/τ τk H S s H m Hk. Note that ou uppe boud holds eve whe the odes ae allowed to appoach abitaily close to each othe, without equiig a lowe boud o iteode distace. This is geeally doe to avoid the divegece of the sigal powe atteuatio fuctio at 0. Howeve, the lowe boud costuctio does ot deped o this divegece. Thus ou bouds hold fo dese as well as expadig etwoks. By cosideig a domai of size A ad shig it to domai of uit size usig the mappig P i A fo the powe levels, we get the followig coollay: Coollay 4.. Fo a ad hoc etwok with odes i a domai with aea A, ude the Physical Model with path loss expoet >, maximum powe level bouded by P max H m N 0 (A, fo ay chael with badwidth Hm ad Noise spectal desity N 0, the taspot capacity scales as Θ( A whe each ode has a powe costait. I paticula, fo etwoks with domai aea Θ(, the taspot capacity gows as Θ(. 5. CAPACITY BOUNDS FOR HYBRID NET- WORKS WITH POWER CONTROL I this sectio, we coside the thoughput capacity of static adom wieless etwoks with a ovelay of adomly placed base statios coected by high thoughput coectios. Ou goal is to show that i additio to povidig a thoughput of Θ( to each ode, oe ca povide a thoughput log of Θ( to a guaateed factio of the odes. This esult shows that the taspot capacity of adom hybid etwoks scales liealy i the umbe of odes. We coside a etwok fomed by wieless odes ad µ base statios, µ > 0, i a disk D of uit aea. Both the ACM SIGCOMM Compute Commuicatios Review 77

8 base statios ad the wieless odes ae assumed to be adomly distibuted uifomly i the disk. The base statios ae assumed to be coected to each othe with a vey high badwidth etwok so that thee ae o bottleecks associated with the base statios. Futhe, the wieless odes ca commuicate with the base statios ad with each othe usig omi-diectioal ateas. Let X k, k ( + µ, deote the locatio of ode k, whee the ode may eithe be a base statio o a wieless ode. We coside both the Potocol Model ad the Physical Model fo tasmissio as defied i Sectio. We will assume that odes ca pefom powe cotol i cotast to [6]. That is, odes ca abitaily choose thei tasmissio adii i the Potocol Model, o thei powe levels i the Physical Model. 5. A Spatial Tessellatio We ow poceed to costuctively show how to povide Θ( thoughput to Θ( odes. We begi by cosideig a spatial tessellatio of the disk. We equie each cell esultig fom the tessellatio to cotai exactly oe wieless ode. Nodes i each cell will be allowed to commuicate oly with a base statio i thei espective cells. A atual tessellatio scheme cosideed hee is the Voooi tessellatio of the disk [9] usig the base statios as geeatos. Biefly, a Voooi tessellatio of a set S with geeato set {a, a,..., a } assigs cell V (a i to poit a i, i, whee V (a i {x S : x a i mi j x a j }. (7 We fist develop some useful popeties of this tessellatio. cell. Thus the fatess of V (a i is at least the aea of C. I geeal, fo ay set G G, the total fatess of all cells coespodig to geeatos i G is at least a k M(a k. (9 4 a k G Suppose ow that thee exist f cells whose total fatess is at most c, ad let the set of coespodig geeatos be B : {b, b,..., b f }. Coside the lagest set S of pais (c k, M(c k, such that c k B ad o two pais i S shae the same poits, i.e., the images ude M ae distict ad the image set is disjoit fom the domai set. Let : S. The f 3. (0 This is because, give that (c k, M(c k is i S, a pai (c j, M(c j will be elimiated oly if M(c k c j (at most oe such pai possible o M(c k M(c j (at most five such pais possible o c k M(c j (at most six such pais possible. Thus at most twelve othe pais will be elimiated. I the followig, let : f. We ca ow boud the pobability i (8 as 3 follows: Defiitio 5.. Call a Voooi cell V (a i, η-fat, if a ope disk of aea η ceteed at a i does ot itesect with the Voooi cell of ay othe geeato. Call the cell η-thi if it is ot η-fat. Let t i : sup{η : η > 0, V (a i is η-fat}, deote the fatess of cell V (a i. Next, we establish a esult about the total fatess of ay set of cells, i.e., the sum of the fatess of each cell i a set of cells. Lemma 5.. Coside the Voooi tessellatio of a uit disk D usig uifom iid poits as geeatos. The, fo 0 < f <, c < f c ad c 3 e( + 4 ( + 4, lim P ob( f cells with total fatess c 0. (8 Poof: Let the set of geeatos be G : {a, a,..., a }, with a i chose adomly i a uifom iid fashio. Let M(a i deote the poit i G\{a i} closest to a i. Note that the fuctio M maps at most 6 poits to the same value. To see why, ote that if M(a i M(a j a k, the we have a i a k a i a j ad a j a k a i a j. Thus a ia k a j, ad cosequetly a 3 k ca seve as the closest poit to at most six othe poits. Next we costuct a lowe boud fo the fatess of a cell i tems of the fuctio M. Note fist that if M(a i a j ad C is a disk ceteed at a i with adius a j a i, the C D V (a i. Hece, C does ot itesect with ay othe P ob( f cells with total fatess c P ob( distict poits c,..., c, d,... d : d i M(c i, i, c k d k 4c k (usig (9 C P P ob( c k d k 4c k (summig ove all ways of choosig pais C P P ob( ρ k δ 4c δ k (whee ρ k : c k d k ad δ > 0 C P P ob( ρ k δ k, k, ad k (,..., 0 k 4c δ C P (,..., 0, k 4c/δ C P ( (,..., 0, k k 4c/δ C P (δ (usig AM-GM ieq. ( (,..., 0, k 4c/δ ρ k P ob( δ k, k ( k + δ + k ACM SIGCOMM Compute Commuicatios Review 78

9 C P (δ ( (,..., 0, k 4c/δ 4c + δ (usig Cauchy ieq., k 4c δ c C P (δ ( + 4 δ k (,..., 0, k 4c/δ c C P (δ ( + 4 δ + 4c δ. ( The last iequality follows fom the fact that the umbe of o-egative itege solutios to c, o equivaletly to c is the coefficiet of x c i ( x +. This coefficiet is +c C c, which is less tha +c. To futhe simplify (, ote that fo sufficietly lage ad 0 < < /, C P!!(! ( e + ( e ( e (usig Stilig s appox. < ( e + ( < ( e + ( ( (whee : < ( e + ( e ( e ( e. ( The last iequality follows fom the fact that fo ay x > 0, e x. To see this, ote that l t t fo t > 0 ad x x set t /x. We ow simplify ( by settig δ c ad usig (: P ob( f cells with total aea c c C P (δ ( + 4 δ + 4c < ( e ( e ( c ( ( 3 4 (ce( e f ( ( e ( c f c 3 f 0 as (ecallig c < f c. 5. Taspot Capacity of Radom Wieless Netwoks with Ifastuctue Suppot δ We will ow demostate how, by employig powe cotol, a costat Θ( thoughput ca be suppoted fo ay factio f, 0 < f <, of odes. We will assume that each souce chooses a destiatio adomly. The outig stategy adopted by each ode will be vey simple. Each ode always tasmits diectly to its closest base statio, choosig its tasmissio age appopiately. The packet the taveses the ifastuctue etwok to each the base statio closest to the destiatio ode. The the base statio adjusts its tasmissio age, ad tasmits the packet ove a wieless hop to its destiatio ode. We begi by showig which odes ca ideed achieve a costat thoughput. Let X i be ay abitay base statio, ad let a k be the geeato of the Voooi cell cotaiig X i. Let be the guad zoe paamete, as defied befoe fo the Potocol Model. Defiitio 5.. Call a base statio X i good if the cell cotaiig X i, V (a k, is (( + X i a k -fat. Call a Voooi cell V (a k itself good if it has at least oe good base statio, else call the cell itself bad. Next we show how the factio of bad cells depeds upo the umbe of base statios ad the umbe of wieless odes. Lemma 5.. Coside a adom etwok with wieless odes ad µ base statios. Suppose 0 < f < ad µ > l γ f c f β, whee γ f :, c (f f ( f f f f c ad β 0.4, (+ with c as defied i Lemma 5.. The lim P ob( f bad cells 0. (3 Poof: Coside a Voooi cell, V (a k, which is bad. Let the fatess of V (a k be t k. Sice the cell is bad, it is possible it has o base statios at all. Now suppose the cell has a base statio X i. Sice V (a k is (( + X i a k -thi, we must have ( + X i a k > t k, o equivaletly, X i a k > t k. Thus X (+ i does ot lie iside a disk t D k ceteed at a k with aea ( k, o equivaletly, disk (+ D k does ot cotai ay base statios. With at least two cells i the domai D, the fatess t k is at most the aea of D ad hece the aea of D k is at most /4 the aea of D. Note that the factio β of D k that must lie iside D is maximized whe D k is the lagest possible, ad is give by β cos ( / > 0.4. Thus we have show that at 4 βt k least a aea ( β (+ t k is devoid of base statios. We ow boud the pobability that thee exist at least f bad cells: P ob( f bad cells P ob( f bad cells whose total fatess < c f + P ob( f bad cells whose total fatess c f. ACM SIGCOMM Compute Commuicatios Review 79

10 Tag limits o both sides, the followig holds: lim sup P ob( f bad cells lim sup P ob( f bad cells whose total fatess < c f + lim sup P ob( f bad cells whose total fatess c f lim sup P ob( f bad cells whose total fatess c f (usig Lemma 5.. (4 Now we uppe boud the ight had side above by usig the fact that fo ay bad cell, at least a aea equal to a factio β of its fatess is devoid of ay base statios. Thus, P ob( f bad cells whose total fatess c f (P ob(cells V (d,..., V (d f have {d,...,d f } total fatess c f P ob(at least aea β c f of the f cells is devoid of base statios Sum of the fatess of the f cells c f ( β c f µ P ob(cells {d,...,d f } V (d,..., V (d f have total fatess c f C f ( β c f µ. We simplify this fo lage by usig the fact that x e x fo all x, ad Thus, C f! (f!(( f! ( e + f( f f ( f( ( f e e ( f ( (f f ( f f (fo sufficietly lage γ f. P ob( f bad cells whose total fatess c f C f ( β c f µ γ f e β c f µ e l(γ f β c f µ 0 as 0. Combiig this with (4 yields the esult. Lemma 5. gives us a lowe boud o the atio of the umbe of base statios to the umbe of wieless odes, to guaatee that o moe tha a cetai factio f of the cells ae bad. It ca be show that each value of µ > 0 coespods to a value of f, 0 < f <. This is because the lowe l γ boud of µ, f c f β f, is a mootoically deceasig fuctio with limits ad 0 espectively at f 0 ad f. This implies that if the atio of wieless odes to base statios is bouded above, oe always has some factio of good cells i the etwok. As we will show, these good cells coespod to wieless odes that ca commuicate with a base statio without distubig ay othe wieless ode. We ae ow eady to state ou mai esult. Theoem 5.. Fo a adom etwok with wieless odes ad µ base statios, ad whee each ode chooses a adom destiatio, it is possible to povide Θ( thoughput log to all the coectios with high pobability. I additio, simultaeously, it is possible to povide a Θ( thoughput to a cetai factio f of wieless odes. Fomally, fo each µ > 0, thee is a f, 0 < f <, such that lim P ob( f odes that ca achieve Θ( thoughput. (5 Poof: We begi by showig that ay two good odes, i.e., odes with good Voooi cells, ca commuicate with thei espective closest base statios simultaeously without itefeig with each othe. Let the two good odes be a i ad a j ad let thei closest base statios be X i ad X j. Let P be ay poit i V (a j. Fom the defiitio of a good ode, we kow that V (a i is ((+ X i a i -fat. Fom the defiitio of η fat, P a i ( + X i a i. Usig the tiagle iequality, P X i P a i X i a i ( + X i a i. Sice s ay poit i V (a j, this shows that the tasmissio betwee X i ad a i is ot distubed by the the tasmissio betwee X j ad a j. This holds fo ay pai of odes. It is easy to see that ay coectio with both the souce ad destiatio as good odes ca be povided Θ( thoughput. Fom Lemma 5., we kow that some costat factio of odes ae good. Thus with high pobability, some f, f > 0, coectios ca be seved Θ( thoughput. It is easy to povide Θ( thoughput to all the othe coectios. log Assumig a slotted opeatio, we ca eseve all the eve slots fo the good coectios selected above. I the odd slots, we ca u aothe policy, like the oe give i [6], that guaatees Θ( thoughput. log 6. CONCLUDING REMARKS We have show that fo the Physical Model of commuicatio, the best case taspot capacity of a ad hoc wieless etwok gows asymptotically as Θ(. The same boud holds eve whe the odes ae allowed to appoach abitaily close to each othe ad eve whe the tasmissio ate is give by the Shao s logaithmic fuctio of the SINR. This closes the gap betwee the peviously kow uppe boud of Θ( ad lowe boud of Θ(, whee > is the path loss expoet. We have also show that the costaits of the Potocol Model ae sticte tha that of the Physical Model, thus allowig diect geealizatios of lowe boud costuctios fo the Potocol Model to the Physical Model. Fo the case of hybid wieless etwoks, we have show how employig powe cotol allows bette scalig of etwok capacity. Specifically, oe ca guaatee a Θ( thoughput to ACM SIGCOMM Compute Commuicatios Review 80

11 some Θ( odes i the etwok. We show how the umbe of such odes depeds upo the umbe of base statios. Ou esult impoves ove pevious esults that establish a maximum thoughput capacity of Θ( pe ode i the log absece of powe cotol. 7. REFERENCES [] P. Gupta ad P. R. Kuma, The capacity of wieless etwoks, IEEE Tasactios o Ifomatio Theoy, vol. IT-46, Mach 000. [] A. Agawal ad P. R. Kuma, Impoved capacity bouds fo wieless etwoks, Wieless Commuicatios ad Mobile Computig, vol. 4, issue 3, pp. 5-6, May 004. [3] L. Xie ad P. R. Kuma, A etwok ifomatio theoy fo wieless commuicatio: Scalig laws ad optimal opeatio, IEEE Tasactios o Ifomatio Theoy, vol. 50, o. 5, pp , May 004. [4] M. Faceschetti, O. Dousse, D. Tse, P. Thia, Closig the gap i the capacity of adom wieless etwoks, pepit. [5] B. Liu, Z. Liu, ad D. Towsley, O the capacity of hybid wieless etwoks, i Poceedigs of IEEE Ifocom, 003. [6] U. C. Kozat ad L. Tassiulas, Thoughput capacity of adom ad hoc etwoks with ifastuctue suppot, i Poceedigs of Mobicom, 003. [7] P. Gupta, Desig ad pefomace aalysis of wieless etwoks, Ph.D. dissetatio, Uivesity of Illiois at Ubaa-Champaig, 000. [8] T. Cove ad J. A. Thomas, Elemets of Ifomatio Theoy. Wiley, 99. [9] A. Okabe, B. Boots, ad K. Sugihaa, Spatial Tessellatios Cocepts ad Applicatios of Voooi Diagams. Joh Wiley, 99. [0] Piyush Gupta ad P. R. Kuma, Iteets i the sky: The capacity of thee dimesioal wieless etwoks, Commuicatios i Ifomatio ad Systems, vol., issue, pp , Ja. 00. [] Piyush Gupta ad P. R. Kuma, Citical powe fo asymptotic coectivity i wieless etwoks, pp , i Stochastic Aalysis, Cotol, Optimizatio ad Applicatios: A Volume i Hoo of W.H. Flemig. Edited by W.M. McEeay, G. Yi, ad Q. Zhag, Bikhause, Bosto, 998. ISBN ACM SIGCOMM Compute Commuicatios Review 8